Noncommutative symmetric functions with matrix parameters

TL;DR

This paper introduces new families of noncommutative symmetric and quasi-symmetric functions parameterized by matrices, unifying and extending existing frameworks like noncommutative Macdonald functions.

Contribution

It defines a generalized class of noncommutative symmetric functions with matrix parameters, connecting various known families through specializations.

Findings

New families of functions depending on matrix parameters

Unification of existing noncommutative symmetric functions

Framework for further generalizations and specializations

Abstract

We define new families of noncommutative symmetric functions and quasi-symmetric functions depending on two matrices of parameters, and more generally on parameters associated with paths in a binary tree. Appropriate specializations of both matrices then give back the two-vector families of Hivert, Lascoux, and Thibon and the noncommutative Macdonald functions of Bergeron and Zabrocki.